The table represents a logarithmic function [tex]f(x)[/tex].

| x | y |
| -------- | --- |
| [tex]\frac{1}{125}[/tex] | -3 |
| [tex]\frac{1}{25}[/tex] | -2 |
| [tex]\frac{1}{5}[/tex] | -1 |
| 1 | 0 |
| 5 | 1 |
| 25 | 2 |
| 125 | 3 |

Use the description and table to graph the function and determine the domain and range of [tex]f(x)[/tex]. Represent the domain and range with inequality notation, interval notation, or set-bulder notation. Explain your reasoning.

Identify the base of the logarithmic function as 5, so f ( x ) = lo g 5 ​ ( x ) .
Determine the domain of f ( x ) as all positive real numbers: 0"> x > 0 or ( 0 , ∞ ) .
Determine the range of f ( x ) as all real numbers: ( − ∞ , ∞ ) .
The domain is 0"> x > 0 and the range is ( − ∞ , ∞ ) , so the final answer is: Domain: 0"> x > 0 , Range: ( − ∞ , ∞ ) . 0, \text{ Range: } (-\infty, \infty)}"> Domain: x > 0 , Range: ( − ∞ , ∞ ) ​

Explanation

Understanding the Problem We are given a table of values for a logarithmic function f ( x ) and asked to determine its domain and range, represent them in different notations, and graph the function.

Finding the Logarithmic Function First, let's identify the base of the logarithmic function. We know that a logarithmic function has the form f ( x ) = lo g b ​ ( x ) , where b is the base. From the table, we can see that when x = 5 , y = 1 . This means lo g b ​ ( 5 ) = 1 . Therefore, the base b must be 5. So, the function is f ( x ) = lo g 5 ​ ( x ) .

Determining the Domain Now, let's determine the domain of the function f ( x ) = lo g 5 ​ ( x ) . The domain of a logarithmic function is all positive real numbers, since we can only take the logarithm of a positive number.

Representing the Domain We can represent the domain in the following ways:



Inequality notation: 0"> x > 0
Interval notation: ( 0 , ∞ )
Set-builder notation: 0\}"> { x ∣ x > 0 }


Determining the Range Next, let's determine the range of the function f ( x ) = lo g 5 ​ ( x ) . The range of a logarithmic function is all real numbers, since the logarithm can take any real value.

Representing the Range We can represent the range in the following ways:



Inequality notation: − ∞ < y < ∞
Interval notation: ( − ∞ , ∞ )
Set-builder notation: { y ∣ y ∈ R }


Graphing the Function Finally, we can plot the points from the table and sketch the graph of the function f ( x ) = lo g 5 ​ ( x ) . The points are ( 125 1 ​ , − 3 ) , ( 25 1 ​ , − 2 ) , ( 5 1 ​ , − 1 ) , ( 1 , 0 ) , ( 5 , 1 ) , ( 25 , 2 ) , and ( 125 , 3 ) . The graph will pass through these points and extend infinitely in both directions.

Final Answer In summary, the logarithmic function is f ( x ) = lo g 5 ​ ( x ) . The domain is 0"> x > 0 , or ( 0 , ∞ ) , or 0\}"> { x ∣ x > 0 } . The range is all real numbers, or ( − ∞ , ∞ ) , or { y ∣ y ∈ R } .


Examples
Logarithmic functions are used in many real-world applications, such as measuring the intensity of earthquakes on the Richter scale, modeling population growth, and calculating the pH of a solution. Understanding the domain and range of logarithmic functions is crucial for interpreting these applications correctly. For example, when measuring earthquake intensity, the magnitude is calculated using a logarithmic scale. The domain ensures that we are only considering positive intensity values, and the range allows us to represent a wide spectrum of earthquake magnitudes, from very small to extremely large.

Answered by GinnyAnswer | 2025-07-03

The function represented is f ( x ) = lo g 5 ​ ( x ) with a domain of 0"> x > 0 and a range of all real numbers ( − ∞ , ∞ ) . All interpretations and representations of the domain and range can be expressed in respective notational formats. Additionally, upon plotting the function, a distinct logarithmic curve will be observed.
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Answered by Anonymous | 2025-07-04